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Bimodal t-Ratios

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Abstract

This paper studies the sampling distribution of the conventional t-ratio when the sample comprises independent draws from a standard Cauchy (0,1) population. It is shown that this distribution displays a striking bimodality for all sample sizes and that the bimodality persists asymptotically. An asymptotic theory is developed in terms of bivariate stable variates and the bimodality is explained by the statistical dependence between the numerator and denominator statistics of the t-ratio. This dependence also persists asymptotically. These results are in contrast to the classical t statistic constructed from a normal population, for which the numerator and denominator statistics are independent and the denominator, when suitably scaled, is a constant asymptotically. Our results are also in contrast to those that are known to apply for multivariate spherical populations. In particular, data from an n dimensional Cauchy population are well known to lead to a t-ratio statistic whose distribution is classical t with n-1 degrees of freedom. In this case the univariate marginals of the population are all standard Cauchy (0,1) but the sample data involves a special form of dependence associated with the multivariate spherical assumption. Our results therefore serve to highlight the effects of the dependence in component variates that is induced by a multivariate spherical population. Some extensions to symmetric stable populations with exponent parameter alpha does not equal 1 are also indicated. Simulation results suggest that the sampling distributions are well approximated by the asymptotic theory even for samples as small as n = 20.

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File URL: http://cowles.econ.yale.edu/P/cd/d08a/d0842.pdf
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Bibliographic Info

Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 842.

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Length: 26 pages
Date of creation: Jul 1987
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Publication status: Published in Econometrics Journal (2010), 13(2): 271-289
Handle: RePEc:cwl:cwldpp:842

Note: CFP 1385.
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  1. Peter C.B. Phillips, 1982. "Exact Small Sample Theory in the Simultaneous Equations Model," Cowles Foundation Discussion Papers 621, Cowles Foundation for Research in Economics, Yale University.
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Cited by:
  1. Peter C.B. Phillips & Mico Loretan, 1989. "The Durbin-Watson Ratio Under Infinite Variance Errors," Cowles Foundation Discussion Papers 898R, Cowles Foundation for Research in Economics, Yale University, revised Aug 1989.
  2. Phillips, P.C.B., 1990. "Time Series Regression With a Unit Root and Infinite-Variance Errors," Econometric Theory, Cambridge University Press, vol. 6(01), pages 44-62, March.
  3. Peter C.B. Phillips, 1987. "Conditional and Unconditional Statistical Independence," Cowles Foundation Discussion Papers 824R, Cowles Foundation for Research in Economics, Yale University, revised Dec 1987.
  4. Peter C.B. Phillips & Mico Loretan, 1990. "Testing Covariance Stationarity Under Moment Condition Failure with an Application to Common Stock Returns," Cowles Foundation Discussion Papers 947, Cowles Foundation for Research in Economics, Yale University.

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